Which shows the correct substitution of the values [tex]\(a, b\)[/tex], and [tex]\(c\)[/tex] from the equation [tex]\(1 = -2x + 3x^2 + 1\)[/tex] into the quadratic formula?

Quadratic formula: [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]

A. [tex]\(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)}\)[/tex]

B. [tex]\(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(2)}}{2(3)}\)[/tex]

C. [tex]\(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(1)}}{2(3)}\)[/tex]

D. [tex]\(x = \frac{-3 \pm \sqrt{3^2 - 4(-2)(0)}}{2(-2)}\)[/tex]



Answer :

Let's proceed step-by-step to determine the correct substitution of the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula from the given quadratic equation.

1. Given Quadratic Equation:
The given quadratic equation is:
[tex]\[ 1 = -2x + 3x^2 + 1 \][/tex]

2. Simplifying the Equation:
Subtracting 1 from both sides, we get:
[tex]\[ 0 = 3x^2 - 2x + 0 \][/tex]
or:
[tex]\[ 3x^2 - 2x = 0 \][/tex]

3. Identifying Coefficients:
We can identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the simplified equation [tex]\(3x^2 - 2x + 0 = 0\)[/tex]:
[tex]\[ a = 3, \quad b = -2, \quad c = 0 \][/tex]

4. Quadratic Formula:
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

5. Substitution into the Quadratic Formula:
Substituting the identified values [tex]\(a = 3\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 0\)[/tex]:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot 0}}{2 \cdot 3} \][/tex]
Simplifying further:
[tex]\[ x = \frac{2 \pm \sqrt{4 - 0}}{6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4}}{6} \][/tex]
[tex]\[ x = \frac{2 \pm 2}{6} \][/tex]

6. Solving for [tex]\(x\)[/tex]:
We get two solutions:
[tex]\[ x_1 = \frac{2 + 2}{6} = \frac{4}{6} = \frac{2}{3} \quad \text{or} \quad 0.6666666666666666 \][/tex]
[tex]\[ x_2 = \frac{2 - 2}{6} = \frac{0}{6} = 0 \][/tex]

Thus, the results are [tex]\(x_1 = 0.6666666666666666\)[/tex] (or [tex]\(\frac{2}{3}\)[/tex]) and [tex]\(x_2 = 0\)[/tex].

The expression that shows the correct substitution of the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)} \][/tex]

So, the correct substitution is:
[tex]\[ \boxed{x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)}} \][/tex]